-
J. Fluid Mech. (2010), vol. 660, pp. 354–395. c© Cambridge
University Press 2010doi:10.1017/S0022112010002818
Zigzag instability of vortex pairs in stratifiedand rotating
fluids. Part 1. General
stability equations.
PAUL BILLANT†LadHyX, CNRS, École Polytechnique, F-91128
Palaiseau Cedex, France
(Received 5 May 2009; revised 12 May 2010; accepted 13 May
2010;
first published online 21 July 2010)
In stratified and rotating fluids, pairs of columnar vertical
vortices are subjectedto three-dimensional bending instabilities
known as the zigzag instability or as thetall-column instability in
the quasi-geostrophic limit. This paper presents a
generalasymptotic theory for these instabilities. The equations
governing the interactionsbetween the strain and the slow bending
waves of each vortex column in stratifiedand rotating fluids are
derived for long vertical wavelength and when the two vorticesare
well separated, i.e. when the radii R of the vortex cores are small
comparedto the vortex separation distance b. These equations have
the same form as thoseobtained for vortex filaments in homogeneous
fluids except that the expressions ofthe mutual-induction and
self-induction functions are different. A key difference isthat the
sign of the self-induction function is reversed compared to
homogeneousfluids when the fluid is strongly stratified: |Ω̂max | N
, the self-induction function is complex becausethe bending waves
are damped by a viscous critical layer at the radial location
wherethe angular velocity of the vortex is equal to the
Brunt–Väisälä frequency.
In contrast to previous theories, which apply only to strongly
stratified non-rotating fluids, the present theory is valid for any
planetary rotation rate and whenthe strain is smaller than the
Brunt–Väisälä frequency: Γ/(2πb2) � N , where Γ isthe vortex
circulation. Since the strain is small, this condition is met
across a widerange of stratification: from weakly to strongly
stratified fluids. The theory is furthergeneralized formally to any
basic flow made of an arbitrary number of vortices instratified and
rotating fluids. Viscous and diffusive effects are also taken into
accountat leading order in Reynolds number when there is no
critical layer. In Part 2 (Billantet al., J. Fluid Mech., 2010,
doi:10.1017/S002211201000282X), the stability of vortexpairs will
be investigated using the present theory and the predictions will
be shown tobe in very good agreement with the results of direct
numerical stability analyses. Theexistence of the zigzag
instability and the distinctive stability properties of vortex
pairsin stratified and rotating fluids compared to homogeneous
fluids will be demonstratedto originate from the sign reversal of
the self-induction function.
Key words: geophysical and geological flows, instability, vortex
flows
† Email address for correspondence:
[email protected]
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Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 355
1. IntroductionA three-dimensional instability, called zigzag
instability or tall-column instability,
has been observed on co-rotating and counter-rotating columnar
vertical vortex pairsin stratified and rotating fluids (Dritschel
& de la Torre Juárez 1996; Billant &Chomaz 2000a; Otheguy,
Billant & Chomaz 2006a; Otheguy, Chomaz & Billant2006b;
Deloncle, Billant & Chomaz 2008; Waite & Smolarkiewicz
2008). The zigzaginstability consists of three-dimensional bending
of the vortices with weak coredeformations. Ultimately, it
generates thin horizontal layers and may explain thelayering
observed in stratified flows (Riley & Lelong 2000) and the
structure ofquasi-geostrophic turbulence (Dritschel, de la Torre
Juárez & Ambaum 1999).
In the case of counter-rotating vortex pairs, the initial
evolution of the zigzaginstability in strongly stratified fluids
(Billant & Chomaz 2000a) qualitativelyresembles that of the
Crow instability in homogeneous fluids (Crow 1970), exceptthat the
Crow instability bends the vortices symmetrically with respect to
the middleplane whereas the zigzag instability is antisymmetric
(Billant & Chomaz 2000a). Inthe case of co-rotating vortex
pairs, the zigzag instability is symmetric (Otheguy et al.2006a ,b)
whereas no long-wavelength bending instability occurs in
homogeneousfluids (Jimenez 1975). In the latter case, only the
elliptic instability has been observedbut such instability is of
different nature since it distorts the vortex core
structure(Meunier & Leweke 2005; Le Dizès 2008).
The Crow instability of counter-rotating vortex pairs in
homogeneous fluid is dueto the interaction between the strain that
each vortex exerts on its companion andthe so-called slow bending
modes of each vortex (Crow 1970; Widnall, Bliss & Zalay1971).
This particular bending mode corresponds to a deflection of the
vortex tubewith negligible internal deformations and is called
‘slow’ because its frequency tendsto zero in the long-wavelength
limit (Leibovich, Brown & Patel 1986).
In order to prove that the same physical mechanism is at work in
the case of thezigzag instability and to provide a complete theory
of the zigzag instability in stratifiedand rotating fluids, the
first step is therefore to theoretically describe the dynamicsof
slow bending waves of a vortex in stratified and rotating fluids in
the presence ofa companion vortex. This is the subject of the
present paper. In a companion paper(Billant et al. 2010)
(hereinafter referred to as Part 2), the stability of vortex pairs
willbe investigated using such theoretical description and the
predictions will be shownto fully explain the existence and
characteristics of the zigzag instability.
In homogeneous fluids, the Crow instability has been described
theoretically byconsidering vortex filaments (Crow 1970). The
vortex filament method is valid forlarge vortex separation and
long-wavelength bending disturbances, and is thereforeparticularly
suited to the Crow instability. This method relies upon the use of
theBiot–Savart law to compute the induced motions and upon the
cutoff approximationto determine the self-induced motion of the
vortices. The latter approximation consistsof integrating the
Biot–Savart law over all of the vortex except a small segment
oneither side of the point where the velocity is evaluated. This
amounts to take intoaccount the finite size of the vortex cores in
order to avoid the logarithmic singularityof the Biot–Savart law.
More fundamentally, the vortex filament method is based onthe
theorems of Helmholtz and Kelvin, which state that vortex lines
move as materiallines and conserve their circulation in homogeneous
and inviscid fluids.
In a stratified and rotating fluid, these theorems are no longer
valid, meaning thatvortex filament method cannot be used.
Nevertheless, the Ertel’s theorem states thatthe potential
vorticity is conserved following the motion. In the
quasi-geostrophiclimit, the potential vorticity and the
streamfunction of the flow are related by a
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356 P. Billant
linear operator relationship so that the horizontal induced
motion is also givenby a Biot–Savart law. One could therefore
consider vortex filaments of potentialvorticity in
quasi-geostrophic fluids and resort to the same method as Crow
(1970).However, a difficulty of this method is that it needs to be
completed by using thecutoff approximation. In homogeneous fluids,
the choice of the cutoff distance hasbeen justified rigorously by
determining the self-induced motion of a single vortexwith a large
curvature by means of matched asymptotic expansions which take
intoaccount the finite size of the vortex core (Widnall et al.
1971; Moore & Saffman1972; Leibovich et al. 1986). However,
such asymptotic analysis is not very far fromdirectly considering
the full problem of two well-separated vortices perturbed
bylong-wavelength bending disturbances. This is the strategy we
have chosen here fora stratified and rotating fluid. A major
advantage is that we will be able to carry outthe analysis for any
Rossby number and over a large range of Froude numbers, i.e.for
conditions much wider than those of the quasi-geostrophic regime.
Furthermore,the analysis will be conducted for any vortex pair of
arbitrary relative strength andany velocity profile of the
individual vortices. Viscous and diffusive effects will alsobe
taken into account at leading order. Thus, we shall extend in many
directionsthe previous theoretical analyses of the zigzag
instability which have been performedonly in strongly stratified
non-rotating inviscid fluids and for the specific cases of
theLamb–Chaplygin counter-rotating vortex pair (Billant &
Chomaz 2000b) and twoequal-strength co-rotating Lamb–Oseen vortices
(Otheguy, Billant & Chomaz 2007).These analyses have shown that
the zigzag instability for these two basic flows canbe interpreted
as a breaking of translational or rotational invariances of the
globalbasic flow for long vertical wavelength in a strongly
stratified fluid. However, such anapproach is difficult to
generalize to weakly stratified-rotating fluids or to other
basicflows. In contrast, the present theory has a general formalism
which enables its usefor any basic flow with an arbitrary number of
vortices.
The paper is organized as follows. We first compute in § 2.2 a
basic state made oftwo vertical vortices when they are well
separated, i.e. when the ratio of separationdistance b to vortex
radius R is large: b/R � 1. After having non-dimensionalizedthe
governing equations in § 2.3, the three-dimensional stability
analysis of this basicstate is analysed asymptotically in § 3 for
long-wavelength bending perturbations.The resulting stability
equations describe the coupling between the strain and theslow
long-wavelength bending disturbances of each vortex. They happen to
have thesame form as those obtained by Crow (1970), Jimenez (1975)
and Bristol et al. (2004)in homogeneous fluids using vortex
filaments except that the explicit forms of themutual-induction and
self-induction functions are different in stratified and
rotatingfluids. The properties of the mutual-induction and
self-induction functions in stratifiedand rotating fluids are
analysed and compared to their counterparts in homogeneousfluids in
§ 4. Finally, the stability equations are generalized to any number
of vorticesin § 5.
2. Stability problem2.1. Governing equations
We consider a rotating, stably stratified fluid under the
Boussinesq approximation.The equations of momentum, continuity and
density conservation read
DûDt̂
+ 2Ωbez × û = −1
ρ0∇p̂ − gρ̂
ρ0ez + ν�û, (2.1)
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Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 357
Γ (l)
θ
r
η
ξ
(0, 0) (b̃, 0)
b̃
x
y
(xc, yc)
Γ (r)
Figure 1. Sketch of the vortex pair in the frame of reference
where it is steady. (x, y) are theCartesian coordinates centred on
the left vortex. (r, θ ) and (ξ, η) are the cylindrical
coordinatescentred on the left and the right vortex, respectively.
The rotation centre of the vortex pair islocated at (xc, yc).
∇ · û = 0, (2.2)Dρ̂
Dt̂+
∂ρ̄
∂ẑûz = D�ρ̂, (2.3)
with û being the velocity, Ωb the rotation rate about the
vertical axis, ez the verticalunit vector, ûz the vertical
velocity, p̂ the pressure, g the gravity, ν the viscosity, �the
Laplacian and D is the diffusivity of the stratifying agent. The
total density fieldρt has been decomposed as ρt (x̂, t̂) = ρ0 +
ρ̄(ẑ) + ρ̂(x̂, t̂), with ρ0 being a constantreference density,
ρ̄(ẑ) a linear mean density profile and ρ̂(x̂, t̂) a perturbation
density.The Brunt–Väisälä frequency N =
√−(g/ρ0)∂ρ̄/∂ẑ, measuring the density gradient,
is assumed to be constant.
2.2. The basic flow
We consider two columnar vertical vortices of circulation Γ (l)
and Γ (r) separated by adistance b in the frame of reference
rotating at rate Ωb (figure 1). The fluid is assumedinviscid and
non-diffusive. When the radii R(l) and R(r) of each vortex are
smallcompared to b, the vortices rotate around each other at rate f
= (Γ (l) + Γ (r))/(2πb2)exactly like two point vortices and each
vortex adapts to the strain field generatedby the other vortex.
Moore & Saffman (1975) (see also Saffman 1992, Rossi 2000and Le
Dizès & Laporte 2002) have shown that this adaptation can be
computedasymptotically when the two vortices are well separated.
For clarity, we briefly repeattheir analysis below.
We first switch from the planetary frame rotating at absolute
angular velocity Ωbto the reference frame rotating at absolute
angular velocity f + Ωb, where the vortexpair is steady. We also
make the problem dimensionless by using the quantities of
the vortex labelled with superscript (l). Time is
non-dimensionalized by 2πR(l)2/Γ (l)
and horizontal length by R(l) (The corresponding non-dimensional
variables will bedenoted without a hat.) The non-dimensional
circulation of the vortex labelled withsuperscript (r) is Γ̃ = Γ
(r)/Γ (l) and its non-dimensional radius R̃ = R(r)/R(l). The
non-dimensional separation distance is b̃ = b/R(l) and the
non-dimensional rate of rotation
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358 P. Billant
of the pair is f̃ , where f̃ = 1 + 1/Γ̃ and
= Γ̃ /b̃2 (2.4)
is the non-dimensional strain.The centre of the vortex labelled
with the superscript (l) is chosen to be located
on the left at (x = 0, y = 0) and the centre of the vortex
labelled with the superscript(r) is on the right at (x = b̃, y =0)
(figure 1). The rotation centre of the vortex pair isat (xc = b̃/f̃
, yc =0). The basic flow Ub can be written in term of a
streamfunction:Ub = −∇ × ψbez , which can be decomposed as
ψb = ψ(l)b + ψ
(r)b + ψf , (2.5)
where ψ (l)b and ψ(r)b are the streamfunctions corresponding to
each vortex and
ψf = −f̃ [(x −xc)2 + (y −yc)2]/2 is the streamfunction of the
solid-body rotation dueto the rotation of the frame of reference
relative to the planetary reference frame.Note that in the limit Γ̃
= −1, we have ψf = b̃x up to an arbitrary constant, meaningthat the
co-moving reference frame no longer rotates but translates along
the y-axis.The streamfunction of each vortex can be decomposed as
follows:
ψ(i)b = ψ
(i)a + ψ
(i)d , (2.6)
for i = {l, r}, where ψ (i)a is the streamfunction of the vortex
(i) as if it were alone andψ
(i)d corresponds to its adaptation to the strain induced by the
other vortex.The ratio between the vorticity of each vortex core
and the strain induced by its
companion is O(1/). Therefore, when the two vortices are well
separated, i.e. b̃ � 1,the streamfunctions ψ (l)d and ψ
(r)d are O() � 1 and can be computed asymptotically.
Let us consider the vortex (l). The condition that the flow is
steady is
J (ψb, �ψb) =∂ψb
∂x
∂�ψb
∂y− ∂ψb
∂y
∂�ψb
∂x= 0, (2.7)
where J denotes the Jacobian. If we focus on the region close to
the core of the leftvortex, this gives at zeroth order in
J(ψ (l)a , �ψ
(l)a
)= 0. (2.8)
This equation is satisfied by any axisymmetric vortex with
streamfunction ψ (l)a (r, θ) ≡ψ (l)a (r), with (r, θ) being the
cylindrical coordinates centred on the left vortex. At firstorder
in , (2.7) gives
J(ψ (l)a , �
(ψ
(l)d + ψf + ψ
(r)b
))+ J
(ψ
(l)d + ψf + ψ
(r)b , �ψ
(l)a
)= 0. (2.9)
This equation can be simplified using the fact that the
streamfunction ψ (r)b of the rightvortex tends to the one of a
point vortex in the neighbourhood of the left vortex (i.e.x − b̃ �
R̃ and y � R̃):
ψ(r)b ∼ ψ (r)a ∼
Γ̃
2ln
((x − b̃)2 + y2
)=
Γ̃
2
[ln b̃2 − 2 r
b̃cos θ − r
2
b̃2cos 2θ + O
(1
b̃3
)],
(2.10)
where we have anticipated that ψ (r)d vanishes outside the core
of the right vortex.
After an integration, (2.9) then leads to the following equation
for ψ (l)d :
∂ψ (l)a∂r
�ψ(l)d −
∂�ψ (l)a∂r
ψ(l)d = −
∂�ψ (l)a∂r
(f̃
2r2 +
1
2r2 cos 2θ
)+ G(r), (2.11)
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Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 359
where G(r) is an arbitrary function. In order that ψ (l)d → 0 as
r → ∞ and benon-singular at r = 0, one has to impose G(r) = f̃
(r2/2)∂�ψ (l)a /∂r . This impliesthat the streamfunction ψ (l)d is
independent of the rotation rate f̃ of the vortexpair in contrast
to the analysis of Le Dizès & Laporte (2002). The reason for
thisdifference is that the latter authors assume ψf = O(1) whereas
ψf = O() in the presentsituation.
The solution of (2.11) is then of the form ψ (l)d = (/2)h(r) cos
2θ , where the functionh can be determined numerically with the
boundary conditions h(r) → 0 as r →∞ and h(r) ∝ r2 as r → 0. As
shown by Moore & Saffman (1975) (see alsoSaffman 1992 and Le
Dizès & Laporte 2002), the streamfunction ψ (l)d correspondsto
an enhancement of the strain in the core of the vortex due to the
interactionbetween strain and vorticity. This is the so-called
internal strain which vanishesrapidly outside the vortex core: h(r)
→ 0 as r → ∞. The latter condition is in factthe only property
needed in the asymptotic analysis of § 3. We shall see that
theexplicit knowledge of the function h(r) is not necessary except
when a critical layerexists.
In conclusion, the basic flow near the vortex labelled (l) can
be written forr � b̃:
Ub =∂ψ (l)a∂r
eθ + U s + O(
1
b̃3
), (2.12)
with U s = −∇ × ψsez , where
ψs = − 12[(r2 − h(r)) cos 2θ + f̃ r2
](2.13)
corresponds to a non-uniform rotating straining flow. Similar
expressions canbe obtained near the right vortex. Finally, far from
the two vortex cores, thestreamfunction ψb of the base flow tends
to the streamfunction of two pointsvortices
ψb =Γ̃
2ln((x − b̃)2 + y2) + 1
2ln(x2 + y2) − f̃
2
[(x − xc)2 + (y − yc)2
]. (2.14)
2.3. Scaling analysis
As mentioned before, the horizontal length unit is taken as the
vortex radius R(l)
of the left vortex and the time unit is chosen as 2πR(l)2/Γ (l).
Accordingly, the
horizontal velocity ûh is non-dimensionalized by Γ (l)/(2πR(l))
and the pressure byρ0Γ
(l)2/(2πR(l))2.Since we shall be mostly interested by stratified
flows for which the horizontal
Froude number,
F(l)h =
|Γ (l)|2πR(l)2N
, (2.15)
will be small, it is convenient to scale the vertical coordinate
by F (l)h R(l), the
vertical velocity ûz by F(l)h Γ
(l)/(2πR(l)) and density fluctuations ρ̂ by
ρ0Γ(l)N/(g2πR(l))
following Billant & Chomaz (2001).
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360 P. Billant
We now write the non-dimensional governing equations in the
reference framerotating at non-dimensional rate f̃ + 1/(2Ro(l))
where the base flow is steady:
∂uh∂t
+ uh · ∇huh + uz∂uh∂z
+
(2f̃ +
1
Ro(l)
)ez × uh = −∇hp +
δΓ
Re(l)�suh, (2.16)
F(l)h
2[∂uz
∂t+ uh · ∇huz + uz
∂uz
∂z
]= −∂p
∂z− ρ + F (l)h
2 δΓ
Re(l)�suz, (2.17)
∇h · uh +∂uz
∂z= 0, (2.18)
∂ρ
∂t+ uh · ∇hρ + uz
∂ρ
∂z= uz +
δΓ
Re(l)Sc�sρ, (2.19)
where the non-dimensional variables are denoted without a hat,
∇h is thehorizontal gradient, �s = �h + F
(l)h
−2∂2/∂z2, with �h being the horizontal Laplacian,
δΓ = sgn(Γ(l)), Sc = ν/D is the Schmidt number and
Ro(l) =Γ (l)
4ΩbπR(l)2, Re(l) =
|Γ (l)|2πν
, (2.20a, b)
are the Rossby number and Reynolds numbers. Note that the
Froude, Rossby andReynolds numbers defined in Otheguy et al. (2006a
,b, 2007) are twice those used inthe present paper. It should be
emphasized that this non-dimensionalization is only aconvenient way
to rewrite the equations for the study of strongly stratified flows
but(2.16)–(2.19) remain valid for any Froude number.
Correspondingly, the Froude number F (r)h , the Rossby number
Ro(r) and the
Reynolds number Re(r) for the right vortex are defined as (2.15)
and (2.20) withthe superscript (l) replaced by (r).
3. Asymptotic three-dimensional stability analysisWe now subject
the basic flow to infinitesimal three-dimensional perturbations
denoted by a tilde:
[u, p, ρ] (x, y, z, t) = [Ub, Pb, 0] (x, y) + Re([ũ, p̃, ρ̃]
(x, y, t)eikz
), (3.1)
where k is the non-dimensional vertical wavenumber and Re
denotes the real part.The corresponding dimensional wavenumber is
k̂ = k/(F (l)h R
(l)), due to the non-dimensionalization of § 2.3. In the main
part of the analysis, we shall considerthe equations of an inviscid
and non-diffusive fluid except if they become singular.Viscous
effects will then be re-introduced into the problem in order to
smooth thesingularity. The viscous and diffusive effects when there
is no singularity will also beconsidered in Appendix D. The
non-dimensional linearized equations (2.16)–(2.19),governing the
disturbances for Re(l) = ∞, are
∂ ũh∂t
+ Ub · ∇hũh + ũh · ∇hUb +(
2f̃ +1
Ro(l)
)ez × ũh = −∇hp̃, (3.2)
F(l)h
2(
∂ũz
∂t+ Ub · ∇hũz
)= −ikp̃ − ρ̃, (3.3)
∇h · ũh + ikũz = 0, (3.4)∂ρ̃
∂t+ Ub · ∇hρ̃ = ũz. (3.5)
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Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 361
It will also be convenient to use the equation for the vertical
vorticity ζ̃ = (∇ × ũh)ez:
∂ζ̃
∂t+ Ub · ∇hζ̃ + ũh · ∇h�ψb − ik
(�ψb +
1
Ro(l)+ 2f̃
)ũz = 0. (3.6)
The stability problem will be solved asymptotically for
well-separated vortices, i.e.a small strain, = Γ̃ /b̃2 � 1, and for
long-wavelength disturbances such that
µ ≡ kmin
(1, |Ro(l)|
)max
(1, F (l)h
) � 1. (3.7)This condition comes from the fact that the leading
three-dimensional terms for a
given Froude number scale as k2 for a large Rossby number and as
k2/Ro(l)2
fora small Rossby number. Alternatively, if the Rossby number is
fixed, the leadingthree-dimensional terms are proportional to k2
for F (l)h
-
362 P. Billant
1
r
∂rũr0
∂r+
1
r
∂ũθ0
∂θ= 0, (3.15)
∂ρ̃0
∂t+ Ω
∂ρ̃0
∂θ= ũz0, (3.16)
where Ω(r) = (1/r)∂ψ (l)a /∂r and ζ (r) = �ψ(l)a are the basic
angular velocity and vertical
vorticity of the left vortex at leading order.The horizontal
velocity of the perturbation obeys purely two-dimensional
equations.
Since the basic flow at leading order is axisymmetric, we can
seek the solution in theform ũh0 = −∇ × (ψ̃0ez) with ψ̃0 =ϕ0(r)
exp(imθ − iω0t). Then, (3.12)–(3.13) give
∂2ϕ0
∂r2+
1
r
∂ϕ0
∂r−
[m2
r2+
ζ ′
r(Ω − ω0)
]ϕ0 = 0, (3.17)
where the prime denotes differentiation with respect to r .
Here, we shall considerbending waves of the vortex, i.e. waves with
azimuthal wavenumbers |m| =1. In thiscase, the general solution of
(3.17) can be found for any angular velocity profile Ω(Michalke
& Timme 1967; Widnall et al. 1971):
ϕ0 = Cr(Ω − ω0) + Dr(Ω − ω0)∫
dr
r3(Ω − ω0)2, (3.18)
where C and D are constants. The second solution is singular at
r = 0 and thereforeone has to set D = 0. The first solution is
non-singular at r = 0 since Ω(0) is assumedto be finite. However,
this solution is unbounded as r → ∞ when ω0 = 0 sinceϕ0 ∼ C(1/r −
ω0r) for r � 1. The matching with a decaying outer solution is
possibleonly if ω0 = 0.
Therefore, the total zero-order streamfunction can be written in
the form
ψ̃0 = rΩ(C
(l)+ (T )e
iθ + C(l)− (T )e−iθ), (3.19)
where C(l)+ and C(l)− are the amplitudes of the waves with
azimuthal wavenumbers
m =1 and m = −1. They are assumed to be function of the slow
time scale T = t .This remarkable solution derives from the
translational invariances. Indeed, (3.19)can be rewritten as
ψ̃0 = −�x(l)∂ψ (l)a∂x
− �y(l) ∂ψ(l)a
∂y, (3.20)
with the relations
�x(l) = −C(l)+ − C(l)− , (3.21)�y(l) = −i
(C
(l)+ − C(l)−
). (3.22)
If we add the infinitesimal perturbation (3.20) to the
streamfunction of the basic flowψ (l)a , we have
ψ (l)a (x, y) + Re(ψ̃0eikz) ∼ ψ (l)a
(x − Re
(�x(l)eikz
), y − Re
(�y(l)eikz
)), (3.23)
meaning that the solution ψ̃0 simply corresponds to a
displacement ofRe(�x(l) exp(ikz)) and Re(�y(l) exp(ikz)) of the
left vortex as a whole in the x andy directions. Since k is assumed
to be small but non-zero, the whole vortex tubeis sinusoidally bent
along the vertical without deformations in the horizontal
plane.Weak radial deformations will, however, be found at the next
orders. These waves aregenerally called ‘slow bending waves’
because their frequency is zero in the limit k = 0
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Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 363
(Leibovich et al. 1986). They are different from other bending
waves which may existfor finite vertical wavenumber or finite
frequency and which have a different radialstructure.
The corresponding vertical vorticity, pressure and vertical
velocity given by (3.12)–(3.16) are
ζ̃0 = ζ′(C(l)+ eiθ + C(l)− e−iθ), (3.24)
p̃0 = rΩ(Ω + Ro(l)
−1)(C
(l)+ e
iθ + C(l)− e−iθ), (3.25)
ũz0 = W+C(l)+ e
iθ − W−C(l)− e−iθ , (3.26)
where
W+ = W− = Wi ≡ Ω2r Ω + Ro
(l)−1
1 − F (l)h2Ω2
. (3.27)
The expression for the amplitude Wi of the vertical velocity is
valid for all r whenF
(l)h < 1/Ωmax , where Ωmax is the maximum non-dimensional
angular velocity of the
left vortex. (Note that we assume that Ω decreases monotonically
with r as observedfor most vortex profiles.) This condition is
equivalent to |Ω̂max |/N < 1, where Ω̂maxis the corresponding
maximum dimensional angular velocity. When F (l)h > 1/Ωmax ,the
vertical velocity amplitude (3.27) presents a singularity at the
radius rc, whereΩ(rc) = 1/F
(l)h . Such singularity can be understood as a resonance when
the local
Doppler shifted frequency of the slow bending mode, i.e. −ω̂0
+mΩ̂ = ± Ω̂ at leadingorder and in dimensional form, is equal to
the Brunt–Väisäilä frequency. A similarsingularity occurs in the
case of a slightly tilted columnar axisymmetric vortex ina
stratified fluid (Boulanger, Meunier & Le Dizès 2007). Indeed,
the inclination ofthe vortex forces a vertical velocity and density
fields similar to those of the presentzero-order perturbation. Near
this singularity, the diffusive effects and the terms oforder O(),
namely the advection of the perturbation by the straining flow U s
andthe evolution of the perturbation on the slow time T , should be
re-incorporated in(3.14) and (3.16), since they are no longer small
compared to the leading-order terms.Nonlinear effects cannot come
into play in the singular region since we are in theframework of a
linear stability analysis. The structure of this critical layer is
analysedin Appendix A following Boulanger et al. (2007). It is
shown that the vertical velocityamplitudes W+ and W− near rc
become
W± = Wv± ≡ ±iRe(l)
1/3ΩcrcπΛ
Ωc + Ro(l)−1
2F (l)h2Ω ′c
Hi(±iΛx±
)+ O
(
Re(l)
1/3), (3.28)
where the subscript c indicates the value taken at rc, Λ =
−sgn(Γ (l))(2Ω ′c/(1+1/Sc))1/3,Hi is the Scorer’s function
(Abramowitz & Stegun 1965; Drazin & Reid 1981; Gil,Segura
& Temme 2002) and
x± = Re(l)1/3
[r − rc −
2Ωc
(rc −
hc
rc
)cos 2θ −
Ω ′c
(f̃ ± i
∂ lnC(l)±∂T
)]. (3.29)
This solution matches the inviscid solution (3.27) away from rc
since Hi(ξ ) ∼ −1/(πξ )for ξ → ∞ with | arg(ξ )| > π/3. Note
that W± are complex conjugates of one anotherwhen ∂ lnC(l)+ /∂T = ∂
lnC
(l)− /∂T = σ and the growth rate σ is purely real.
-
364 P. Billant
rc
W+
r0 0.5 1.0 1.5
–10
0
10
(a) (b)
∫r ∞W
+(η
)dη
r0.5 1.0 1.5
–2.0
–1.5
–1.0
–0.5
0
0.5
1.0
Figure 2. Vertical-velocity amplitude W+ (a) and integral∫ r
∞ W+(η) dη (b) as a function ofr for Fh = 1.25, Ro = ∞, Re = 50
000, Sc = 1 and = 0 for the Lamb–Oseen vortex (4.7). Thereal and
imaginary parts are shown by solid and dashed lines, respectively.
In (a), the thinlines show the inviscid formula (3.27) and the bold
lines show the viscous solution (3.28). Thedotted line shows the
location of the singularity rc = 0.681. In (b), the thin lines have
beenplotted by integrating the inviscid formula (3.27) in the
complex plane according to the rule(3.38). The bold lines show the
result of the integration of the composite formula (3.30) on
thereal axis.
Figure 2(a) illustrates one example of the velocity amplitude W+
when a criticalpoint exists. The inviscid solution (3.27) is
represented by thin lines and the viscoussolution (3.28) for Re(l)
= 50 000 and = 0 is shown by bold lines. We can see that theviscous
solution smoothes the singularity and perfectly matches the
inviscid solutionaway from the critical radius. It is noteworthy
that the viscous solution has animaginary part (bold dashed line in
figure 2a) in contrast to the inviscid solution.
The critical-layer solution (3.28) is similar to the purely
viscous solution derived byBoulanger et al. (2007) except that two
additional features are taken into account:the elliptical shape of
the vortex and the slow evolution of C(l)± (the last term
of(3.29)). Since these two effects are of order O() whereas viscous
effect scales as
Re(l)1/3
, two different regimes are possible depending upon the number
Re(l)3. WhenRe(l)3 � 1, i.e. for moderate Reynolds number or very
small strain, the terms O()are negligible. The typical amplitude of
the vertical velocity in the critical layer
then scales as Re(l)1/3
and the typical size of the critical layer is Re(l)−1/3
, as foundby Boulanger et al. (2007). In contrast, for higher
Reynolds number, Re(l)3 � 1, theterms O() cannot be neglected. The
solution (3.28) shows that the critical layer is thenconcentrated
around the elliptic streamline whose mean radius is r = rc.
Furthermore,if C(l)+ (respectively C
(l)− ) has a growth rate with a real part of the same sign as
Γ
(r) sothat the dimensional growth rate is positive, the point x±
= 0 is located in the lower(respectively upper) half complex
r-plane (since Ω ′c < 0) at a distance O(||) from thereal
r-axis. Thus, we have |Λx±| � 1 with | arg(±iΛx±)| > π/3 along
the real r-axis sothat the Scorer’s function in (3.28) can be
approximated by Hi(±iΛx±) ∼ i/(±πΛx±).This implies that the typical
amplitude of W± then scales as 1/|| and the typicalwidth of the
critical layer is ||.
A composite approximation uniformly valid in r can be
constructed from (3.27)and (3.28):
W± = Wi + Wv± + Ωcrc
Ωc + Ro(l)−1
2F (l)h2Ω ′c(r − rc)
. (3.30)
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Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 365
In the following, we shall use this composite approximation when
a critical pointexists.
3.1.2. Order- problem
At order , the horizontal momentum equations can be written in a
compact andconvenient form in terms of an equation for the the
first-order vertical vorticity ζ̃1:
Ω∂ζ̃1
∂θ+ ũr1ζ
′ = − ∂ζ̃0∂T︸︷︷︸(a)
− U s · ∇ζ̃0 − ũh0 · ∇�ψs︸ ︷︷ ︸(b)
+ ik2
(1
Ro(l)+ ζ
)ũz0︸ ︷︷ ︸
(c)
, (3.31)
and the divergence equation is
1
r
∂rũr1
∂r+
1
r
∂ũθ1
∂θ+ i
k2
ũz0 = 0. (3.32)
As seen in (3.31), the first-order vertical-vorticity
perturbation ζ̃1 is forced by threedifferent terms. The term (a)
corresponds to the evolution of the zeroth-ordervertical-vorticity
perturbation on the slow time scale T . The term (b) describesthe
advection of the zeroth-order vertical-vorticity perturbation ζ̃0
by the strainingflow U s and, conversely, the advection of the
vertical vorticity �ψs of the strainingflow by the zeroth-order
velocity perturbation ũh0. The last term (c) represents
thestretching of the vertical vorticity of the basic flow by the
zeroth-order vertical-velocityperturbation ũz0. The latter
vertical velocity also appears as a forcing in the
divergenceequation (3.32). These terms correspond to the leading
three-dimensional effects andare included at this order due to the
assumption µ2 = O().
Note that the viscous diffusion of ζ̃1 is always negligible at
leading order in Reynoldsnumber even when a critical point rc
exists. Therefore, the viscous effects do not needto be considered
in (3.31) and are present only implicitly through the vertical
velocityũz0 when there is a critical point.
In order to solve (3.31) and (3.32), we decompose the horizontal
velocity intorotational and potential components with a
streamfunction ψ̃1 and a potential Φ̃1:
ũh1 = −∇ × (ψ̃1ez) + ∇hΦ̃1. (3.33)Using (3.26), the divergence
equation (3.32) becomes
�hΦ̃1 = −ik2
(W+C
(l)+ e
iθ − W−C(l)− e−iθ). (3.34)
The solution is found by reduction of order to be
Φ̃1 = −ik2
(H+(r)C(l)+ eiθ − H−(r)C(l)− e−iθ
), (3.35)
with
H±(r) =r
2
∫ r∞
W±(η) dη −1
2r
∫ r0
η2W±(η) dη, (3.36)
where the limits of integration have been chosen so that Φ̃1 is
not singular at r =0and vanishes as r → ∞ for finite Froude number
Fh.
The functions W± in (3.36) are given by the inviscid expression
(3.27) when
F(l)h < 1/Ωmax and by the composite approximation (3.30)
otherwise. However, as
usual for viscous critical layers (Lin 1955; Drazin & Reid
1981; Huerre & Rossi1998; Le Dizès 2004), the effect of the
critical layer can be more simply taken
-
366 P. Billant
into account by using the inviscid function (3.27) for all
radius but by bypassing thesingularity in the complex plane. This
equivalence is based on the fact that the integralof Scorer’s
function verifies
∫ xHi(ξ ) dξ ∼ − ln x/π for x → ∞ with | arg(x)| > π/3.
Thus, when integrating, for example, the viscous function Wv+
given by (3.28) fromone side of the critical point to the other
side, we have∫ rc+δr
rc−δrWv+ dr ∼ −iΩcrcπ
Ωc + Ro(l)−1
2F (l)h2Ω ′c
sgn(Γ (l)Ω ′c
), (3.37)
where δr is much larger than the typical width of the critical
layer. The sameresult is obtained by integrating the inviscid
function W+ along a path, avoiding thesingularity in the upper
(respectively lower) half complex plane when sgn(Γ (l)Ω ′c)
isnegative (respectively positive). Since Ω ′c < 0 for most
angular velocity profiles, thefollowing rule should be adopted when
computing the function with subscript + in(3.36):
the contour of integration has to be deformed in the upper half
complex plane
when the dimensional circulation of the vortex is positive: Γ
(l) > 0 and in the
lower half when Γ (l) < 0. (3.38)
The rule is reversed for the function with subscript −.An
example of this equivalence is illustrated in figure 2(b), which
shows the
first integral of (3.36), i.e.∫ r
∞ W+(η) dη. The bold lines represent the result of
theintegration of the composite approximation (3.30) while the thin
lines are the resultof the integration of the inviscid solution
(3.27) in the complex plane. We see that bothmethods give the same
result outside the critical layer and lead, in particular, to
thesame phase jump when crossing rc. However, it should be stressed
that the amplitudesof the variations inside the critical layer are
arbitrary for the contour-deformationmethod and depends directly on
the radius of the small semicircular detour used toavoid the
singularity.
By introducing the decomposition (3.33) in (3.31), we obtain an
equation for thestreamfunction ψ̃1:
Ω∂�hψ̃1
∂θ− 1
r
∂ψ̃1
∂θζ ′ = −Us · ∇ζ̃0 − ũh0 · ∇�ψs︸ ︷︷ ︸
(b)
−∂ζ̃0∂T
+ ik2
(1
Ro(l)+ ζ
)ũz0 −
∂Φ̃1
∂rζ ′.
(3.39)
Using (3.19) and (3.24), the straining terms, denoted by (b) in
(3.39), can be rewrittenas
(b) = J
(ψs, �x
(l) ∂�ψ(l)a
∂x+ �y(l)
∂�ψ (l)a∂y
)+ J
(�x(l)
∂ψ (l)a∂x
+ �y(l)∂ψ (l)a∂y
, �ψs
).
(3.40)
From (2.9), the streamfunction of the straining flow ψs
satisfies
J(ψs, �ψ
(l)a
)+ J
(ψ (l)a , �ψs
)= 0. (3.41)
By deriving this equation with respect to x and y, we find
that
(b) = Ω∂�hψ̃1s
∂θ− 1
r
∂ψ̃1s
∂θζ ′ with ψ̃1s = −�x(l)
∂ψs
∂x− �y(l) ∂ψs
∂y. (3.42)
Therefore, the solution forced by the straining terms in (3.39)
is ψ̃1s and simplycorresponds to a displacement by an amount equal
to (�x(l), �y(l)) of the straining flow
-
Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 367
ψs like the leading-order perturbation (3.20). This means that
the total perturbationψ̃0 + ψ̃1s consists of a displacement of the
elliptic vortex as a whole. Using (2.13)and the relations
(3.21)–(3.22), the streamfunction ψ̃1s can be explicitly written
as
ψ̃1s = −f̃ r(C
(l)+ e
iθ + C(l)− e−iθ) + (h′(r)
4+
h(r)
2r− r
)(C(l)− e
iθ + C(l)+ e−iθ)
+
(h′(r)
4− h(r)
2r
)(C
(l)+ e
3iθ + C(l)− e−3iθ). (3.43)
The last two terms of the right-hand side of (3.39) correspond
to three-dimensionaleffects. The corresponding solution can be
obtained by reduction of order. Therefore,the whole solution of
(3.39) can be found analytically:
ψ̃1 = ψ̃1s − ir(
∂C(l)+
∂Teiθ − ∂C
(l)−
∂Te−iθ
)
+k2
[ (F+(r) − rH′+(r)
)C
(l)+ e
iθ +(F−(r) − rH′−(r)
)C(l)− e
−iθ], (3.44)where
F±(r) = rΩ(r)[∫ r
0
W±(η)
Ω(η)dη +
∫ r0
dη
η3Ω2(η)
∫ η0
(Ω(ξ ) +
1
Ro(l)
)ξ 2W±(ξ )dξ
].
(3.45)
As before, the inviscid expression (3.27) of W± can be used for
all r when F(l)h > 1/Ωmax
provided that the contour of integration is deformed in the
complex plane around thesingularity rc according to the rule (3.38)
for the function with subscript + and thereversed rule for the
function with subscript −.
The first-order solution generally behaves like ψ̃1 ∝ r for
large r . Note that thereis also a term proportional to r ln r in
F±(r), as shown in Appendix B but this doesnot substantially modify
the present reasoning. Since the leading-order inner solutionψ̃0
behaves as 1/r for large r , this implies that the inner solution
ψ̃ = ψ̃0 + ψ̃1 + · · ·is not uniformly asymptotic for all r but
valid only close to the core of the left vortexsuch that r �
min(b̃, 1/µ). In the latter inequality, the two parameters b̃ and
1/µ areconsidered separately even if they are assumed to be
formally of the same magnitude.The purpose is to encompass the case
of a single vortex for which b̃ → ∞ but µ isnon-zero. The goal of
the next subsection is to find a solution valid for large
radius.
3.2. Outer region
We now assume that we are far from the cores of the two vortices
such that r = O(d),where d is an arbitrary large distance: d � 1.
By introducing a long-range variableu = r/d , the basic flow
outside the vortex cores given by (2.14) becomes
Ubr = −Γ̃
b̃
(b̃2
d2u2 − 2db̃u cos θ + b̃2− 1
)sin θ, (3.46)
Ubθ =1
du+ Γ̃
(du − b̃ cos θ
d2u2 − 2db̃u cos θ + b̃2
)+
Γ̃
b̃cos θ − f̃ du. (3.47)
The region u ≈ b̃/d and θ ≈ 0 is excluded since it corresponds
to the inner region ofthe right vortex where the perturbation has
to be calculated in the same way as in theprevious subsection. We
see that the order of magnitude of the base flow for u = O(1)
-
368 P. Billant
is Ub = O(dU ) where
U = max(||, 1/d2) (3.48)is a small parameter.
In order to find the solution in this region, it is convenient
to rescale the linearizedgoverning equations (3.2)–(3.5) with the
long-range radius u:
∂ ũh∂T
+ U(Ūb · ∇̄hũh + ũh · ∇̄hŪb
)+
(2f̃ +
1
Ro(l)
)ez × ũh = −∇̄hΠ̃, (3.49)
F(l)h
2(
∂ũz
∂T+ U Ūb · ∇̄hũz
)= −ikdΠ̃ − ρ̃, (3.50)
∇̄h · ũh + ikdũz = 0, (3.51)
∂ρ̃
∂T+ U Ūb · ∇̄hρ̃ = ũz, (3.52)
where the basic flow and the pressure have been rescaled: Ūb =
Ub/(dU ) = O(1),Π̃ = p̃/d and the horizontal gradient is with
respect to the stretched coordinates:∇̄h = d∇h. The equation for
the vertical vorticity ζ̃ = (∇̄ × ũh)ez becomes
∂ζ̃
∂T+ U Ūb · ∇̄hζ̃ − i
kd
Ro(l)ũz = 0, (3.53)
where we have used the fact that the vertical vorticity of the
basic flow (3.46)–(3.47)is uniform: (∇ × Ub)ez = −2f̃ . Using
(3.52), (3.53) leads to the equation for theconservation of
potential vorticity:
∂q̃
∂T+ U Ūb · ∇̄hq̃ = 0, where q̃ = ζ̃ − i
kd
Ro(l)ρ̃. (3.54a, b)
Since the potential vorticity of the perturbation is initially
zero outside the vortexcores and is advected by the base flow like
a passive tracer, it is legitimate to assumethat it will remain
zero for all time: q̃ = 0, i.e.
ζ̃ = ikd
Ro(l)ρ̃. (3.55)
We also note that (3.52) implies that the vertical velocity ũz
is one order ofmagnitude in U smaller than ρ̃. Thus, (3.50) reduces
to the hydrostatic balance whenF
(l)h U � 1:
−ikdΠ̃ − ρ̃ = 0 + O((
F(l)h U
)2). (3.56)
The parameter kd appearing in (3.56) and (3.51) will be assumed
arbitrary since k issmall but d is large. In this case, the outer
perturbation can be expanded with thesmall parameter U :
ũh = ũh0 + U ũh1 + · · · , (3.57)Π̃ = Π̃0 + UΠ̃1 + · · · ,
(3.58)ũz = U ũz1 + · · · , (3.59)ρ̃ = ρ̃0 + U ρ̃1 + · · · .
(3.60)
Two different cases need to be considered according to the
magnitude of the Rossbynumber Ro(l): Ro(l) � O(1/U ) and Ro(l) �
O(1/U ).
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Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 369
3.2.1. Outer solution for Ro(l) � O(1/U )Inserting the
asymptotic series (3.57)–(3.60) in (3.49)–(3.51) yields at leading
order
1
Ro(l)ez × ũh0 = −∇̄hΠ̃0, (3.61)
−ikdΠ̃0 − ρ̃0 = 0, (3.62)∇̄h · ũh0 = 0, (3.63)
while (3.55) is
ζ̃0 = ikd
Ro(l)ρ̃0. (3.64)
The horizontal velocity at leading order is therefore given
by
ũh0 = −∇̄ × (ψ̃0/dez) + ∇̄hΦ̃0/d, (3.65)
with
�̄hΦ̃0 = 0 and �̄hψ̃0 =
(kd
Ro(l)
)2ψ̃0. (3.66a, b)
These equations are those of a quasi-geostrophic flow. This is
not surprising since weare far from the two vortex cores where the
base flow (3.46)–(3.47) is small. Thus, thelocal Rossby number
Ro(l)U and the Froude number F
(l)h U based on the magnitude
of the base flow in this region are both small even if the
Rossby number Ro(l) andthe Froude number F (l)h based on the
characteristics of the vortex core are large.
The solutions of (3.66b) which decay at infinity are of the form
Km(βdu)e±imθ , i.e.
expressed in terms of the unstretched radius: Km(βr)e±imθ ,
where Km is the modified
Bessel function of the second kind of order m and β = k/|Ro(l)|
=2k̂R(l)|Ωb|/N . Thesolutions of (3.66a) which decay at infinity
are simply of the form r−me±imθ . In orderto be consistent with the
inner solution of each vortex, the streamfunction is taken asthe
superposition of two solutions with azimuthal wavenumbers |m| =1:
one centredon the left vortex and the other centred on the right
vortex:
ψ̃0 = β[K1(βr)
(E
(l)+ e
iθ + E(l)− e−iθ) + Γ̃ K1(βξ )(E(r)+ eiη + E(r)− e−iη)],
(3.67)
where (ξ, η) are the cylindrical coordinates centred on the
right vortex (figure 1) andE
(l)± , E
(r)± are constants that will be later related to the
displacements of each vortex
due to the matching between the inner and outer solutions. The
additional factors βand Γ̃ have been included in (3.67) in order to
simplify the matching procedure. Notethat the potential Φ̃0 should
also be chosen so as to match the 1/r behaviour of theinner
potential Φ̃1 for large r .
3.2.2. Outer solution for Ro(l) � O(1/U )
In this case, the leading-order equations are identical to the
previous ones exceptthat the horizontal momentum equation (3.61)
becomes
0 = −∇̄hΠ̃0. (3.68)
Thus, we have Π̃0 = 0 leading to
�̄hΦ̃0 = 0 and �̄hψ̃0 = 0. (3.69a, b)
Equations (3.69) are those of a two-dimensional potential flow.
The solution whichdecays at infinity and will match the inner
solutions of each vortex can be written in
-
370 P. Billant
the form
ψ̃0 =1
r
(E
(l)+ e
iθ + E(l)− e−iθ) + Γ̃
ξ
(E
(r)+ e
iη + E(r)− e−iη). (3.70)
We can notice that setting 1/Ro(l) � O(U ) in (3.67) also gives
the same solution atthe order considered herein. It is therefore
correct to use (3.67) for all the values ofthe Rossby number.
In contrast, solution (3.67) is restricted to the regime of
small and moderate Froudenumber: F (l)h � 1/U in order that the
hydrostatic balance (3.56) is satisfied. Thisimplies that the
following three conditions should hold:
F(l)h �
1
|| , F(l)h �
1
µ2, F
(l)h � r2. (3.71a, b, c)
In dimensional form, the first condition (i.e. (3.71a)) is Γ
(r)/(2πb2) � N , i.e. thedimensional strain exerted by the
companion vortex should be much smaller than theBrunt–Väisäilä
frequency. Since the strain is typically small, this condition is
fulfilledover a wide range of Froude number. Condition (3.71b) is
equivalent to (3.71a)when µ2 = O(). However, when the separation
distance is very large, b̃ � 1/µ, and,in particular, in the case of
a single vortex (i.e. b̃ = ∞), (3.71b), together with theassumption
(3.7), implies that the solution (3.67) is valid only in the
wavenumberrange 0 � k � min(1, |Ro(l)|)max(1,
√F
(l)h ). Finally, the last condition (i.e. (3.71c)),
combined with the assumption r � 1 made at the beginning of §
3.2, shows that (3.67)is valid only for large radius such that r �
max(
√F
(l)h ,1).
3.3. Matching
The inner solution is valid for r � min(b̃, 1/µ) while the outer
solution (3.67) isvalid for r � max(
√F
(l)h ,1). Therefore, these two solutions should match in the
overlap
region:
max
(√F
(l)h , 1
)� r � min(b̃, 1/µ). (3.72)
Such range exists due to the assumptions (3.71a,b) and (3.7). We
first express the outersolution (3.67) in this overlap region. The
cylindrical coordinates (ξ, η) appearing in(3.67) can be expressed
in terms of (r, θ) using the relations r cos θ − b̃ = ξ cos η andr
sin θ = ξ sin η. Since r � b̃ in the overlap region, we haveξ =
b̃
[1 − rb̃−1 cos θ + O
(r2/b̃2
)]and eiη = −1 + irb̃−1 sin θ + O
(r2/b̃2
). (3.73)
Therefore, the outer solution (3.67) at leading orders
becomes
ψ̃out =
[1
r+
β2r
2
(ln
(βr
2
)− 1
2+ γe
)] (E
(l)+ e
iθ + E(l)− e−iθ)
+
r
2
[Ψ
(E
(r)+ − E(r)−
)(eiθ − e−iθ ) − χ
(E
(r)+ + E
(r)−
)(eiθ + e−iθ )
]+ O(β4r3 ln(βr), r2/b̃3, /r), (3.74)
up to an arbitrary constant and where γe =0.5772 . . . is
Euler’s constant. The functionsχ and Ψ are given by
χ(β̂b) = β̂bK1(β̂b) + β̂2b2K0(β̂b), Ψ (β̂b) = β̂bK1(β̂b),
(3.75a, b)
where
β̂ = β/R(l) = 2k̂|Ωb|/N (3.76)
-
Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 371
is a rescaled dimensional vertical wavenumber independent of the
characteristics ofeach vortex. These two functions will be later
seen to be the equivalent in stratifiedand rotating fluids of the
first and second mutual-induction functions of Crow (1970)in
homogeneous fluids. The function Ψ and χ simply describe an
advection of theleft vortex along the x and y axes, respectively,
by the horizontal velocity of thedisturbance of the right
vortex.
Note that we shall consider that the terms O(β2 lnβ) in (3.74)
are formally of thesame order as those O(β2) in order to avoid the
definition of an additional slowtimescale O(β2 lnβ). This
simplifies the presentation and does not detract from therigour of
the analysis as long as higher-order terms are not computed.
The main difficulty in obtaining the expression of the inner
solution in the overlapregion is determining the behaviour of the
functions F±(r) for r � max(
√F
(l)h ,1). This
analysis is carried out in Appendix B. From the final result (B
10), the expression ofthe inner solution ψ̃in = ψ̃0 + ψ̃1 + O(
2) can be obtained easily, where ψ̃0 is givenby (3.19) and ψ̃1
by (3.44):
ψ̃in =1
r
(C
(l)+ e
iθ + C(l)− e−iθ) − r
[C(l)− e
iθ + C(l)+ e−iθ + i
∂C(l)+
∂Teiθ − i∂C
(l)−
∂Te−iθ
]
+ r
[β2
2
(δ(F
(l)h , Ro
(l))
− 12
+ ln r
)− f̃
]C
(l)+ e
iθ
+ r
[β2
2
(δ∗
(F
(l)h , Ro
(l))
− 12
+ ln r
)− f̃
]C(l)− e
−iθ + O(
r, 2
), (3.77)
where the asterisk denotes the complex conjugate and
δ(Fh, Ro) = D(Fh) + 2RoB(Fh) + Ro2A(Fh). (3.78)
The parameters (A, B, D) are constants depending on the Froude
number given by(B 6)–(B 8) in Appendix B. Incidentally, it is worth
noting that the function h hasdisappeared in (3.77) since h → 0 as
r → ∞. This means that the adaptation of thevortex to the strain
(i.e. the elliptic deformation of the vortex) will play no
activerole in the instability in contrast to the case of the
elliptic instability (Moore &Saffman 1975; Le Dizès &
Laporte 2002). Here, the whole elliptic vortex tube issimply
displaced, as demonstrated by (3.20) and (3.42).
There is no difficulty in matching the outer potential Φ̃0 to
the inner potential Φ̃1for large r since they both behave like 1/r
. In contrast, the two streamfunctions (3.74)and (3.77) match only
if we impose
E(l)+ = C
(l)+ , E
(l)− = C
(l)− , (3.79)
and
∂C(l)+
∂T= iC(l)− + i
(f̃ − ω
(l)
)C
(l)+ +
i
2
[Ψ
(E
(r)+ − E(r)−
)− χ
(E
(r)+ + E
(r)−
)], (3.80)
∂C(l)−
∂T= −iC(l)+ − i
(f̃ − ω
(l)∗
)C(l)− +
i
2
[Ψ
(E
(r)+ − E(r)−
)+ χ
(E
(r)+ + E
(r)−
)], (3.81)
-
372 P. Billant
which are the evolution equations governing C(l)+ and C(l)− over
the slow time T . The
parameter ω(l) is given by
ω(l) ≡ ω(β̂R(l), F
(l)h , Ro
(l))
=
(β̂R(l)
)22
[− ln
(β̂R(l)
2
)+ δ
(F
(l)h , Ro
(l))
− γe
]. (3.82)
The inner perturbation of the right vortex can be determined by
means of ananalysis identical to the one performed previously. In
particular, this analysis leadsto relations identical to (3.79)
with the superscript (l) replaced by (r) and where theamplitudes
C(r)± are related to the displacement perturbations of the right
vortex by
�x(r) = −C(r)+ − C(r)− and �y(r) = −i(C
(r)+ − C(r)−
). (3.83)
We now rewrite (3.80)–(3.81) in terms of these displacement
quantities and thoseof the left vortex defined in (3.21)–(3.22). In
addition, we rescale the slow time T = t
in terms of the dimensional time t̂ = t2πR(l)2/Γ (l). This
gives
∂�x(l)
∂t̂= − Γ
(r)
2πb2�y(l) +
Γ (r)
2πb2Ψ �y(r) +
(f − Γ
(l)
2πR(l)2ω(l)r
)�y(l) +
|Γ (l)|2πR(l)2
ω(l)i �x
(l),
(3.84)
∂�y(l)
∂t̂= − Γ
(r)
2πb2�x(l) +
Γ (r)
2πb2χ�x(r) −
(f − Γ
(l)
2πR(l)2ω(l)r
)�x(l) +
|Γ (l)|2πR(l)2
ω(l)i �y
(l),
(3.85)
where ω(l)r = Re(ω(l)) and ω(l)i =Im(ω
(l)). The corresponding equations for thedisplacement
perturbations of the right vortex have the same form with
thesuperscripts (r) and (l) interchanged
∂�x(r)
∂t̂= − Γ
(l)
2πb2�y(r) +
Γ (l)
2πb2Ψ �y(l) +
(f − Γ
(r)
2πR(r)2ω(r)r
)�y(r) +
|Γ (r)|2πR(r)2
ω(r)i �x
(r),
(3.86)
∂�y(r)
∂t̂= − Γ
(l)
2πb2�x(r)︸ ︷︷ ︸
(a)
+Γ (l)
2πb2χ�x(l)︸ ︷︷ ︸(b)
−(
f︸︷︷︸(c)
− Γ(r)
2πR(r)2ω(r)r
)�x(r) +
|Γ (r)|2πR(r)2
ω(r)i �y
(r)
︸ ︷︷ ︸(d)
,
(3.87)
where ω(r) =ω(β̂R(r), F (r)h , Ro(r)).
Equations (3.84)–(3.87) have exactly the same form as the
stability equationsobtained by Crow (1970), Jimenez (1975) and
Bristol et al. (2004) for a pair of vortexfilaments in homogeneous
fluids except that the mutual-induction functions Ψ and χand the
self-induction function ω are different. These equations combine
the followingfour different physical effects labelled (a–d) in
(3.87):
(a) Strain: This term represents the effect of the strain field
from one vortex actingon the bending perturbations of the other
vortex. For example, if the right vortex isdisplaced towards its
companion (i.e. �x(r) < 0), it starts to move along the y-axis
(see(3.87)) since the advection by the left vortex is then slightly
larger.
(b) Mutual induction: These terms describe the effects of the
bending perturbationsof one vortex on its companion. These effects
depend on the mutual-inductionfunctions Ψ (β̂b) and χ(β̂b), which
describe how the perturbation decays outside
-
Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 373
the vortex core and is thus felt by the companion vortex. They
are independent ofthe characteristics of each vortex. Looking
again, for example, at (3.87), the term (b)simply shows that if the
left vortex is displaced towards the right vortex (i.e. �x(l) >
0),the right vortex starts to move along the y-axis since the
relative advection by theleft vortex is then higher.
(c) This is an orbital rotation effect due to the rotation of
the basic vortex pair.(d) Self-induction: This effect represents
the dynamics of each vortex as if it were
alone. It makes each sinusoidally bent vortex to rotate rigidly
about its own axis atangular velocity ωrΓ/(2πR
2) (we lighten notation by dropping the superscripts (l) or(r)
in the following) in the same direction as the flow in the vortex
core if ωr > 0 andin the opposite direction if ωr < 0. When a
viscous critical layer is present (i.e. ωi = 0),the bending
deformations decrease in time with the damping rate |ωiΓ |/(2πR2).
Forsimplicity, the damping terms have been written in terms of the
absolute value of thecirculations. This way, the contour of
integration near the singularity should alwaysbe deformed in the
upper complex plane whatever the sign of Γ when computing
theparameters A, B and D.
We recall that these equations are valid for well-separated
vortices b � R and longvertical wavelength: k̂FhR � max(1,
√Fh)min(1, |Ro|) (see (3.7) and (3.71b)). The self-
induction and mutual-induction functions derived here for
stratified-rotating fluidsare valid regardless of the Rossby
numbers and when the strains are much smallerthan the
Brunt–Väisäilä frequency, i.e. Γ/(2πb2) � N (see (3.71a)). Since
b � R, theseconditions are fully satisfied over a wide range of
Froude number: Fh � b2/R2.
As shown in Appendix D, viscous and diffusive effects can also
be taken intoaccount at leading order in Reynolds number when there
is no critical layer, i.e. forFh < 1/Ωmax . In this regime, the
self-induction function (3.82) is to be simply replacedby
ω → ω − i k̂2R2
ReV, (3.88)
where V is a constant defined in (D 22), and it depends on the
vortex profile andthe numbers (Fh, Ro, Sc). It is worth emphasizing
that this constant is not alwayspositive, i.e. the viscous effects
are not always stabilizing.
In the next section, we investigate the behaviours of the
self-induction and mutual-induction functions in stratified and
rotating inviscid fluids compared to homogeneousfluids. The study
of the stability of vortex pairs in stratified and rotating fluids
using(3.84)–(3.87) is postponed to Part 2.
4. Self-induction and mutual-induction functions in stratified
and rotatingfluids
The expressions of the mutual-induction and self-induction
functions in stratified-rotating inviscid fluids and homogeneous
fluids are summarized in table 1. Note thatthe self-induction
function ω is not defined as in Crow (1970). Here, the
self-inductionfunction ω multiplied by Γ/(2πR2) exactly corresponds
to the frequency of rotationof the sinusoidally bent vortex whereas
the self-induction function ωc of Crow (1970)is ωc = −ω/(k̂2R2)
(note the minus sign).
As seen in table 1, these functions in stratified–rotating
fluids are different fromtheir counterpart in homogeneous fluids.
The mutual-induction functions dependon β̂b =2bk̂|Ωb|/N in
stratified–rotating fluids instead of k̂b in homogeneous
fluids.Furthermore, the expressions of χ and Ψ are inverted. These
differences come from
-
374 P. Billant
Function Stratified and rotating fluids Homogeneous fluids
χ β̂bK1(β̂b) + β̂2b2K0(β̂b) k̂bK1(k̂b)
Ψ β̂bK1(β̂b) k̂bK1(k̂b) + k̂2b2K0(k̂b)
ωβ̂2R2
2
[− ln
(β̂R
2
)+ δ(Fh, Ro) − γe
]k̂2R2
2
(ln
k̂R
2+ γe − D
)
Table 1. Expressions of the first and second mutual-induction
functions χ and Ψ andself-induction function ω in
stratified–rotating fluids and homogeneous fluids (Crow
1970; Widnall et al. 1971). The parameter k̂ is the dimensional
wavenumber and
β̂ = k̂Fh/|Ro| = 2k̂|Ωb|/N . The parameter δ(Fh,Ro) is defined
in (3.78) and D = D(0) is definedin (B 8). Note that the
self-induction function is not defined as in Crow (1970).
the fact that the perturbation outside the vortex core is
irrotational in homogeneousfluids whereas the governing equation
for the outer perturbation in stratified-rotatingfluids states that
the potential vorticity is zero (see (3.66b)). The
mutual-inductionfunctions are equal to unity for β̂b = 0 and
decrease exponentially for large β̂b. Theyare essentially
negligible for β̂b � 8. Note that in the case of a stratified
non-rotatingfluid (Ωb = 0), we have χ =Ψ = 1 independently of k̂
and b. The only differencebetween the displacement equations
(3.84)–(3.87) and those for two-dimensional pointvortices is then
the presence of self-induction effects.
4.1. General property of the self-induction function
In order to investigate the properties of the self-induction
function in stratified–rotating fluids, it is convenient to rewrite
(3.82) as
ω =k2
2Ro2
[− ln
(k
2|Ro|
)+ D(Fh) + 2RoB(Fh) + Ro2A(Fh) − γe
], (4.1)
where we recall that k = k̂FhR. Some general results can be
first deduced from (4.1)without specifying the vortex profile. When
Fh < 1/Ωmax , the self-induction functionis purely real because
the parameters (A, B, D) are real. For Ro = ∞, the logarithmicterm
(first term on the right-hand side of (4.1)) is dominant for
sufficiently small k,implying that ω is always positive in this
limit. However, the self-induction function ωis higher for cyclonic
vortices (Ro > 0) than for anticyclonic vortices (Ro < 0)
becausethe third term on the right-hand side of (4.1) depends on
the sign of the Rossbynumber and B is positive whatever the vortex
profile (see (B 7)). When Ro = ∞, theself-induction reduces to ω =
k2A/2. As seen from (B 6), A is positive for Fh < 1/Ωmaxso that
the self-induction is also always positive. When Fh > 1/Ωmax ,
the self-inductionbecomes complex with a negative imaginary part
and the sign of the real part of(A, B, D) depends on the Froude
number.
In summary, the key result is that the self-induction function
(4.1) is positive whenFh < 1/Ωmax whatever the vortex profile
and the Rossby number. Physically, thismeans that a sinusoidally
curved vortex spins rigidly about its axis in the same direc-tion
as the rotation of flow in the vortex core. Strikingly, this is
opposite to the case ofa homogeneous fluid where the self-induction
(as defined herein) is negative for k̂ � 1:
ω =k̂2R2
2(γe − D − ln 2 + ln k̂R) + · · · , (4.2)
where the constant D is the same as (B 8) for Fh = 0, i.e. D =
D(0). This formula hasbeen obtained by Widnall et al. (1971) and
Leibovich et al. (1986) (see also Moore &
-
Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 375
xx′
x
dl
Γ
zy
(a) (b)
xx′
dl
Γ
x
zy
Figure 3. Graphical interpretation of the Biot–Savart law in
homogeneous fluid (a) andquasi-geostrophic fluid (b). See
explanations in text.
Saffman 1972) by means of a long-wavelength matched asymptotic
analysis similarto the present one.
The direction of the self-induced motion can be easily
understood in the cases ofhomogeneous fluids and quasi-geostrophic
fluids. Indeed, in both cases, the motioninduced by a vortex line
can be derived from the Biot–Savart law. In homogeneousfluids, the
velocity induced at a point x by all the portions dl(x ′) of a
vortex filamentis
u(x) = − Γ4π
∫(x − x ′) × dl
|x − x ′|3 (4.3)
It can be shown that the dominant contribution in this integral
comes from theportions of the filament in the neighbourhood of x
(Batchelor 1967). The cutoffapproximation should, however, be used
(i.e. the immediate neighbourhood of xshould be excluded) to avoid
the logarithmic singularity (Widnall et al. 1971).Figure 3(a)
graphically illustrates the velocity induced at point x by a nearby
portionof a curved vortex filament. Due to the curvature of the
vortex, the point x is locatedto the left of the line tangent at x
′ (i.e. parallel to dl). This directly shows that theinduced motion
is directed in the negative y direction. The motion induced by
theother portions of the filament is also in the same direction.
Thus, the self-inducedmotion tends to rotate the curved vortex in a
direction opposite to the flow in thevortex core.
In the case of a quasi-geostrophic fluid, the induced motion is
given by a similarlaw:
u(x̃) = − Γ4π
∫(x̃ − x̃ ′) × dlez
|x̃ − x̃ ′|3 , (4.4)
where x̃ = xex + yey + z̃ez, where z̃ =(2Ωb/N)z is the rescaled
vertical coordinate(see, for example, Miyazaki, Shimada &
Takahashi 2000). The crucial difference isthat the vector dl = dlez
always remains vertical even when the filament is curvedsince the
motion is constrained to remain horizontal. As sketched in figure
3(b), thepoint x is now located to the right of the vertical line
going through x ′ (i.e. parallelto dl). Therefore, the induced
motion is directed in the positive y direction so thatthe curved
vortex will spin in the same direction as the flow in the vortex
core.
-
376 P. Billant
The sign of the self-induction function can also be easily
understood in the case ofa stratified non-rotating fluid. In this
case, we have ω = k2A/2 and the parameter Acan be written in terms
of the leading-order vertical-velocity amplitude W+:
A =∫ ∞
0
ξ 2Ω(ξ )W+(ξ ) dξ. (4.5)
Thus, we see that the real part of A will be positive when the
vertical-velocityamplitude W+ is positive over all ξ . The fact
that a positive vertical-velocity amplitudeinduces a positive
self-induced motion is explained in Otheguy et al. (2007) forFh = 0
(see figure 3b of Otheguy et al. 2007). For example, if we assume a
positivedisplacement of the vortex in the y direction (i.e. �y >
0 and �x = 0), the divergenceequation at leading order is
∇ · ũh = k2�yW+ cos θ. (4.6)
This shows that the vertical-velocity field generates a
divergence for x = r cos θ < 0and a convergence for x > 0
when W+ > 0. To satisfy mass conservation, a secondaryhorizontal
motion is thus created in the negative x direction. This tends to
rotate thevortex in the same direction as the flow in the vortex
core. The vertical-velocity fieldalso stretches and squeezes the
basic vorticity so as to displace the vortex in the samedirection
(see Otheguy et al. 2007).
From the expression of the vertical-velocity amplitude (3.27),
we now see that theexplanations of Otheguy et al. (2007) are valid
not only for Fh = 0 but for all theFroude numbers in the range Fh
< 1/Ωmax , since the vertical-velocity amplitude W+is then
positive for all radius. For Fh > 1/Ωmax , W+ remains positive
for r > rc butbecomes negative for r < rc. The motions
described above are reversed in the latterregion. The net effect
(i.e. the sign of the real part of A) will depend on the
relativeimportance of the two regions. Nevertheless, since the size
of the region where W+ isnegative increases with the Froude number,
the net self-induced motion is expectedto reverse for large Froude
number.
4.2. Self-induction function of the Lamb–Oseen vortex
In order to provide more quantitative results on the
self-induction in stratified–rotatingfluids, we now consider the
Lamb–Oseen vortex profile
Ω =1
r2(1 − e−r2 ), (4.7)
The parameters A, B and D corresponding to this profile have
been computednumerically and are shown in figure 4. Note that they
can be computed analyticallyfor Fh = 0:
A(0) = 9 ln 3 − 14 ln 2 = 0.1834, B(0) = 32(2 ln 2 − ln 3) =
0.4315,
D(0) = γe − ln 22
= −0.0579.
⎫⎬⎭ (4.8)
The value of A(0) agrees with the asymptotic results obtained by
Otheguy et al.(2007) for a strongly stratified non-rotating fluid.
As can be seen in figure 4, theparameters (A, B, D) are almost
constant from Fh = 0 to Fh ≈ 0.5. Their real partshave a peak
around Fh =1/Ωmax = 1 and for Fh > 1, they have a negative
imaginary
-
Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 377
Fh0 1 2 3 4 5
Fh0 1 2 3 4 5
–1.0
–0.5
0
0.5
1.0(a) (b)
–1.0
–0.5
0
0.5
Figure 4. Real part (a) and imaginary part (b) of the parameters
A (——), B (– – –) andD (– · –) as a function of the Froude number
Fh for the Lamb–Oseen vortex (4.7). The dottedline in (a) indicates
the Froude number Fh =1.83 for which the real part of A
becomesnegative.
part. For large Froude number, they behave like
A(Fh) = −1
2F 2h
[lnFh + γe − ln 2 + i
π
2
]+ O
(F −4h
),
B(Fh) =1
2F 2h− i π
4Fh+ O
(F −4h
),
D(Fh) = −1
2ln Fh − i
π
4+ O
(F −4h
),
⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭
(4.9)
so that A and B decrease to zero while D behaves like the
logarithm of the Froudenumber (figure 4). The approximation (4.9)
almost gives the exact values as soon asFh � 4. The parameters (A,
B, D) in the simple case of the Rankine vortex have asimilar
behaviour, as shown in Appendix C.
Figure 5 shows some examples of the self-induction function of
the Lamb–Oseenvortex (represented by thin solid lines) for Ro = ∞
and various Froude numbers fromFh =0.9 to Fh =10. We can see that
the self-induction has a large negative imaginarypart when Fh >
1. It can also be noticed that the real part of the self-induction
ispositive for Fh = 1.67 and negative for Fh = 2. This is because
the sign of the realpart of A changes at Fh = 1.83 (figure 4).
These results have been checked by directly computing the
frequency of the bendingwaves of the Lamb–Oseen vortex (4.7) in a
stratified-rotating fluid. When there is asingle axisymmetric
vortex, the perturbations defined in (3.1) can be further writtenin
the form
[ũ, p̃, ρ̃](r, θ, t) = [û, p̂, ρ̂](r)e−iωet+imθ . (4.10)
With our definition of the self-induction function ω, the
frequency ωe of the slowbending wave with azimuthal wavenumber m
=+1 should be equal to ω in the limitof small vertical
wavenumber.
A single equation for ϕ = rûr can be obtained by inserting
(4.10) in (3.2)–(3.5):(ϕ′
rQ
)′−
[m2
r2− k2 s
2 − φ1 − F 2h s2
+m
r2s
(rζ ′ − (ζ + Ro−1)
(rQ′
Q+ 2
))]ϕ
rQ= 0, (4.11)
where s = −ωe + mΩ , Q = −k2s2/(1 − s2F 2h ) + m2/r2 and φ =(2Ω
+ Ro−1)(ζ + Ro−1).This equation has been solved by a shooting
method similar to the one described by
-
378 P. Billant
ω2π
R2 /Γ
0
0 0 1 2 3 40.2 0.4 0.6 0.8
00.2 0.4 0.6
0
0.05
0.10(a) (b)
(c) (d)
–0.10
–0.05
0
0.05
0.10
–0.10
–0.05
0
–0.10
–0.05
0
ω2π
R2 /Γ
kRFh kRFh
0.2 0.4 0.6
Figure 5. Comparison between the self-induction (real part: thin
solid line; imaginary part:thin dashed line) for a Lamb–Oseen
vortex in a stratified non-rotating fluid and the exactfrequency ωe
of the slow bending mode m= +1 obtained numerically by a shooting
method(real part: thick solid line; imaginary part: thick dashed
line) for various Froude numbers: (a)Fh = 0.9, (b) Fh = 1.67, (c)
Fh = 2 and (d) Fh = 10.
Sipp & Jacquin (2003). The boundary condition at r → ∞ is
ûr → 0 or waves radiateoutwards. At r =0, the boundary condition
is ∂ûr/∂r = 0 since m =1. Singular pointsare bypassed in the
complex plane with the rule (3.38).
The results are shown in figure 5 by bold lines. As expected, we
see that theagreement with the self-induction function ω is
excellent for small wavenumber k.There is a departure as k
increases since ω is only an accurate approximation of ωeat order
O(k2).
When the Froude number becomes large and Ro = ∞, one could
expect that theself-induction function (4.1) tends to that for
homogeneous fluids (cf. (4.2)). However,using (4.9), we see that
for large Froude number Fh and for Ro = ∞, (4.1) tends to
ω =k2A
2≈ − k̂
2R2
4
(ln Fh + γe − ln 2 + i
π
2
). (4.12)
This shows that the real part of the frequency becomes more and
more negativeas Fh increases while the imaginary part is
independent of Fh when representedas a function of k̂R. Therefore,
the self-induction (4.1) is always different from(4.2) even when Fh
→ ∞. In order to resolve this paradox, one has to rememberthat
(4.1) is valid only for small wavenumbers satisfying the condition
(3.71b), i.e.k̂RFh � max(1,
√Fh) for Ro = ∞. Thus, the domain of validity of (4.1) for Fh
> 1,
-
Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 379
0 0.1 0.2 0.3 0.4–0.08
–0.06
–0.04
–0.02
0
ω2π
R2 /Γ
kR
Figure 6. Comparison between the self-induction (real part: thin
solid line; imaginary part:thin dashed line) for a Lamb–Oseen
vortex in a stratified non-rotating fluid for Fh = 50 andthe
frequency ωe of the slow bending mode m= +1 obtained numerically by
a shootingmethod (real part: thick solid line; imaginary part:
thick dashed line). The self-induction inhomogeneous fluids (Fh =
∞) is also shown by a dashed–dotted line. The dotted line showsthe
critical frequency ω = −1/Fh.
i.e. k̂R � 1/√
Fh, shrinks to zero when Fh → ∞. This is equivalent to |ω| �
1/Fh.Figure 6 shows this situation for Fh = 50. (Note that ω and ωe
are plotted as afunction of k̂R instead of k̂FhR as before.) We see
that the frequency ωe obtainednumerically (bold lines) is
discontinuous and exhibits two separate branches, the firstfor
−1/Fh = −0.02 < ωe < 0, which has a negative imaginary part,
and the secondfor ωe < −0.02, which presents a purely real
frequency. The self-induction function(4.1) gives a good
approximation of the first branch whereas (4.2) asymptotes
thesecond branch. There is therefore no contradiction between (4.1)
and (4.2); theysimply apply to two distinct ranges of frequencies:
|ω| � 1/Fh and 1/Fh � |ω| � 1,respectively.
Figure 7 shows some further comparisons for a fixed Froude
number, Fh =0.5,but for various Rossby numbers. The agreement
between the self-induction ω andthe exact frequency ωe is always
good for low wavenumbers. The upper limit of thisrange depends on
the Rossby number. For larger wavenumbers, the
self-inductionfunction (4.1) tends to generally underestimate the
exact frequency of the bendingwave for negative Ro and overestimate
it for positive Ro. We can also clearly seethat the self-induction
function for |Ro| =O(1) is higher for cyclonic vortices (Ro >
0)than for anticyclonic vortices (Ro < 0).
5. Generalization to a basic state with an arbitrary number of
vorticesThe generalization of the asymptotic stability analysis of
§ 3 to a basic flow made
of an arbitrary number N of well-separated vortices presents no
particular difficultyand is straightforward from the analysis of
two vortices. We assume here the existenceof a two-dimensional
vortex configuration which is steady in a given reference frame.Our
purpose is only to derive the general form of the stability
equations for suchbasic flow.
-
380 P. Billant
ω2π
R2 /Γ
0 0.2 0.4 0.6
0
0.05
0.10(a)ω
2πR
2 /Γ
0.20.1 0.3 0.40
0.05
0.10
(c)
kRFh
0.20.1 0.3 0.40
0.05
0.10
(d)
kRFh
0 0.2 0.4 0.6
0
0 0
0.05
0.10(b)
Figure 7. Comparison between the self-induction (thin solid
line) for a Lamb–Oseen vortexin a strongly stratified (Fh =0.5) and
rotating fluid and the exact frequency ωe of the slowbending mode
m= +1 obtained numerically by a shooting method (real part: solid
thick line;imaginary part: thick dashed line) for various Rossby
numbers: (a) Ro = −2.5, (b) Ro = 2.5,(c) Ro = −1.25 and (d) Ro =
1.25.
The approach is the same as carried out in § 3: we consider the
vicinity of agiven vortex (labelled q) and perform an asymptotic
expansion with the small strainsand small vertical wavenumber. The
basic flow near the vortex q (with length non-
dimensionalized by R(q) and time by 2πR(q)2/Γ (q)) has the same
form as (2.12) but
with the strain due to each companion vortex p = {1, 2, . . . ,
N} (p = q):
ψs = −1
̂
∑p =q
Γ (p)
4πb(qp)2(r2 − h(r)
)cos(2θ − 2α(qp)) − f
2̂r2, (5.1)
where b(qp) is the dimensional norm and α(qp) the angle of the
vector joining thecentre of the vortex q to the centre of vortex p
and f is the dimensional angularvelocity of the reference frame in
which the vortex array is steady. The parameter
̂ is now the typical order of magnitude of the dimensional
strains Γ (p)/(2πb(qp)2)
exerted by each vortex p. Accordingly, the small non-dimensional
strain parameter is
= ̂2πR(q)2/Γ (q).
The solution of the zeroth-order inner problem of the stability
analysis is identicalto (3.19). The solution of the first-order
inner problem also has the same form as
-
Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 381
(3.44) with ψ̃1s still given by (3.42), i.e.
ψ̃1s =f r
2̂
((�x(q) − i�y(q))eiθ + (�x(q) + i�y(q))e−iθ
)−
∑p =q
Γ (p)
4πb(qp)2̂
[(h′
4+
h
2r− r
)
×((�x(q) + i�y(q))eiθ−2iα
(qp)
+ (�x(q) − i�y(q))e−iθ+2iα(qp))
+
(h′
4− h
2r
)((�x(q) − i�y(q))e3iθ−2iα(qp) + (�x(q) +
i�y(q))e−3iθ+2iα(qp)
)], (5.2)
where (�x(q), �y(q)) are the displacements of the vortex q . The
outer streamfunction(3.67) becomes
ψ̃0 = β
[K1 (βr)
(E
(q)+ e
iθ + E(q)− e−iθ) + ∑
p =q
Γ (p)
Γ (q)K1
(βξ (p)
) (E
(p)+ e
iη(p) + E(p)− e−iη(p))] ,
(5.3)
where β = k̂F (q)h R(q)/|Ro(q)|, (ξ (p), η(p)) are the
cylindrical coordinates centred on the
vortex p and E(p)± are constants.The matching of the outer and
inner streamfunctions then yields the following
equations for the displacements of the vortex q:
∂
∂t̂
(�x(q)
�y(q)
)= −
∑p =q
Γ (p)
2πb(qp)2R(qp)
(0 11 0
) [R(qp)
]−1 (�x(q)�y(q)
)
+∑p =q
Γ (p)
2πb(qp)2R(qp)
(0 Ψ (β̂b(qp))
χ(β̂b(qp)) 0
)[R(qp)
]−1 (�x(p)�y(p)
)
+
(f − Γ
(q)
2πR(q)2ω(q)r
) (0 1
−1 0
)(�x(q)
�y(q)
)+
|Γ (q)|2πR(q)2
ω(q)i
(�x(q)
�y(q)
),
(5.4)
where ω(q) = ω(β̂R(q), F (q)h , Ro(q)) and
R(qp) =
(cos α(qp) − sinα(qp)
sin α(qp) cos α(qp)
), (5.5)
is the rotation matrix. Even if it takes into account the
effects of several vortices,(5.4) is similar to (3.84)–(3.85) for
the case of two vortices. The only difference isthe presence of the
rotation matrix R(qp) in (5.4) due to the fact that the
Cartesianreference frame is no longer such that the x-axis is
aligned along the line joining thevortices q and p as in
(3.84)–(3.85). However, it can be checked that (5.4) for α(qp) =
0and N =2 reduces to (3.84)–(3.85). As before, these equations have
the same formas those that would be obtained using vortex filaments
with the Biot–Savart law andthe cutoff approximation in homogeneous
fluids.
6. ConclusionIn this paper, we have derived a general
theoretical approach to investigate
the three-dimensional stability of well-separated vertical
columnar vortices with
-
382 P. Billant
respect to long-wavelength bending disturbances in stratified
and rotating fluids.Such approach can be seen as the equivalent of
the stability analyses basedon vortex filaments in homogeneous
fluids (Crow 1970; Jimenez 1975; Bristolet al. 2004). While the use
of vortex filaments is legitimate in homogeneousfluids due to the
Helmholtz and Kelvin theorems, such a concept is not validin
stratified and rotating fluids because the circulation of an
individual vortex isnot conserved. We have therefore derived the
stability equations directly from theEuler equations under the
Boussinesq approximation by resorting to a matchedasymptotic
expansion for well-separated vortices (R/b � 1) and small rescaled
verticalwavenumber k̂RFh � max(1,
√Fh)min(1, |Ro|). Technically speaking, such asymptotic
approach is as accurate as the vortex filament methods in
homogeneous fluids:the stability equations are valid up to orders
O(R2/b2) and O(k̂2R2) in bothcases.
The stability equations have been derived in detail in the case
of two interactingvortices and generalized to the case of several
vortices. The equations are formallyidentical to those in
homogeneous fluids. They express the fact that the
bendingdeformation of a vortex column generally evolves because of
three effects: its ownself-induced motion as if the other vortices
were absent, the strain field generated bythe companion vortices
and the remote effect of their bending disturbances
(mutualinduction).
The nature of the fluid, stratified–rotating or homogeneous,
appears only in themutual-induction and self-induction functions.
We have derived their expressionsfor any Rossby number Ro and for a
wide range of Froude number such thatFh � b2/R2, i.e. when the
strain is much smaller than the Brunt–Väisälä frequency.Compared
to homogeneous fluids, the most crucial difference is the sign of
the self-induction function: it is positive in stratified and
rotating fluids when Fh < 1/Ωmaxwhile it is negative in
homogeneous fluids. Furthermore, the self-induction functionbecomes
complex when Fh > 1/Ωmax because the bending disturbances are
dampedby a viscous critical layer at the radius where the angular
velocity of each vortexis equal to the Brunt–Väisälä frequency.
Dissipative effects can also be taken intoaccount when there is no
critical layer.
The present theory extends in many directions the previous
theoretical analyses ofthe zigzag instability that have been
performed only in strongly stratified non-rotatinginviscid fluids
and for specific basic flows (Billant & Chomaz 2000b; Otheguy
et al.2007). The study of the stability of vortex pairs in
stratified and rotating fluids usingthe final equations
(3.84)–(3.87) is therefore of high interest and should provide
acomprehensive understanding of the zigzag instability. Such
stability analysis willbe carried out in Part 2. The theoretical
predictions will be validated against theresults of direct
numerical stability analyses of co- and counter-rotating vortex
pairs.A very good agreement will be found except for co-rotating
vortex pairs when theRossby number has O(1) negative values. The
discrepancy comes from the fact thatthe self-induction is very low
for these Rossby number values (see figure 7) so thatthe balance
between strain and self-induction effects is actually achieved only
when thevertical wavenumber is no longer small. An improved theory
taking into accounthigher-order three-dimensional effects is
developed in Appendix E for this particularrange of the Rossby
number.
The present theory will also provide a general framework to
understand the physicalmechanism of the zigzag instability. In
particular, the sign reversal of the self-inductionfunction will be
shown to explain why vortex pairs are subjected to the
zigzaginstability in stratified and rotating fluids.
-
Zigzag instability of vortex pairs in stratified and rotating
fluids. Part 1. 383
I would like to thank J. M. Chomaz, A. Deloncle, F. Gallaire, L.
Hua, S. Leblanc, S.Le Dizès, P. Otheguy and the referees for their
helpful comments and suggestions. I’mvery grateful to P. Brancher
and A. Antkowiak for providing their viscous Chebyshevspectral
stability code.
Appendix A. Critical layerIn this appendix, we derive the
vertical velocity of the zeroth-order problem
(§ 3.1.1) near the singular radius rc where Ω(rc) = 1/F (l)h .
Near rc, the neglected terms,namely the advection by the straining
flow, the slow evolution and diffusion of theperturbation are no
longer small compared to the leading-order terms. In order
todetermine the structure of the critical layer, these higher-order
effects therefore needto be re-inserted in (3.14) and (3.16) for
the vertical velocity ũz0 and density ρ̃0:
F(l)h
2P (ũz0) = −ip̃0 − ρ̃0 +δΓ
Re(l)
(F
(l)h
2�hũz0 − k2ũz0
), (A 1)
P (ρ̃0) = ũz0 +δΓ
Re(l)Sc
(�hρ̃0 −
k2
F(l)h
2ρ̃0
), (A 2)
where p̃0 is given by (3.25) and the operator P is
P = Ω ∂∂θ
+
∂
∂T+ U s · ∇. (A 3)
For large Reynolds numbers as considered herein, the terms of
order are generallymuch larger than diffusive effects. Therefore,
at first sight, one could think that onlythe O() terms would need
to be considered near the singularity. However, theseterms cannot
generally prevent the existence of the singularity. Therefore, both
O()terms and viscous terms need to be taken into account. To this
end, the operator Pcan be first simplified near the critical radius
rc by using new coordinates:
s = r −
2Ωc
(rc −
hc
rc
)cos 2θ − rc1, (A 4)
α = θ +
2Ωc
[(1 − h
′c
2rc
)− Ω
′c
2Ωc
(rc −
hc
rc
)]sin 2θ, (A 5)
where the subscript c indicates the value taken at rc. The
variable s is constant alongthe streamline of the basic state whose
mean radius is rc. When the angular velocity Ωis constant, these
coordinates become equivalent to the elliptico-cylindrical
coordinates(Waleffe 1990; Mason & Kerswell 1999) for small
ellipticity. The parameter rc1 hasbeen introduced in (A 4) in order
to anticipate for a slight shift of the critical radius.With these
coordinates, the operator P becomes
P = (Ωc + Ω ′c(s − rc) + (Ω ′crc1 − f̃ ))∂
∂α+
∂
∂T+ O((s − rc), (s − rc)2). (A 6)
Note that the present analysis assumes Ω ′c = 0. It is therefore
not valid when rc isvery close to the vortex axis, i.e. when F (l)h
is just slightly above the critical value1/Ωmax . Following
classical analysis of viscous critical layer, the viscous effects
aretaken into account by introducing the local variable
x = Re(l)1/3
(s − rc), (A 7)
-
384 P. Billant
and expanding the local solution in power of Re(l)1/3
:
(ũz0, ρ̃0) = Re(l)1/3(ũz00, ρ̃00) + (ũz01, ρ̃01) + · · · . (A
8)
At leading order, (A 1)–(A 2) become
F(l)h
2Ωc
∂ũz00
∂α= −ρ̃00, (A 9)
Ωc∂ρ̃00
∂α= ũz00. (A 10)
Writing the vertical velocity in the form
ũz00 = W̃+C(l)+ e
iα − W̃−C(l)− e−iα (A 11)
gives the condition of existence of the critical layer: F
(l)h2Ω2c = 1. At order Re
(l)1/3,we have
F(l)h
2[(Ω ′crc1 − f̃ )